TTF DataScience, WP Statistics
This session is about how statistics can be in service of science.
marginaleffectstargetsIn this session I will use:
marginaleffectstidyversetargetsmgcv (tangentially)knitr (for rendering outputs)Quarto markdown and code here (optional):
I earned my PhD in Evolutionary Anthropology during the 433rd lunation of the third millennium, when Mars was at an orbital longitude of ~311 degrees in May 2022
To get from Zurich to here, I travelled 807 light-microseconds 242 kilometers
The units we use to communicate matter!
‘Monolingual participants have a mean cognitive flexibility score of 51.27 (SD = 18.13), while multilingual participants have a mean score of 61.77 (SD = 18.78).’
marginaleffects package by Vincent Arel-Bundock makes these calculations easy for a wide range of models in both R and Python.
Call:
lm(formula = cognitive_flexibility ~ multilingual)
Residuals:
Min 1Q Median 3Q Max
-52.387 -11.913 1.581 13.388 36.985
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 51.267 1.665 30.795 < 2e-16 ***
multilingual 10.508 2.336 4.499 1.05e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 18.46 on 248 degrees of freedom
Multiple R-squared: 0.07545, Adjusted R-squared: 0.07172
F-statistic: 20.24 on 1 and 248 DF, p-value: 1.051e-05
\[ \text{Cognitive flexibility} = \alpha + \beta \times \text{Multilingual} + \epsilon \] \[ \epsilon \sim \text{Normal}(0, \sigma^2) \]
For a given child, let \(Y_i(\text{multilingual})\) be the outcome if the child is multilingual and \(Y_i(\text{monolingual})\) be the outcome if the child is monolingual.
The estimand is the average difference in potential outcomes: \(\frac{1}{N} \sum_{i=1}^N [Y_i(\text{multilingual}) - Y_i(\text{monolingual})]\). Known as the average treatment effect (ATE), or average marginal effect (AME).
In our simple linear regression, all children are identical, conditional on their mono/multilingual status, which allows us to reduce the estimand to a single parameter.
Call:
lm(formula = cognitive_flexibility ~ multilingual * age_c)
Residuals:
Min 1Q Median 3Q Max
-45.937 -9.753 1.457 12.291 38.200
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 51.123 1.580 32.353 < 2e-16 ***
multilingual 10.969 2.217 4.948 1.39e-06 ***
age_c 2.316 1.079 2.145 0.0329 *
multilingual:age_c 2.949 1.506 1.959 0.0512 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 17.51 on 246 degrees of freedom
Multiple R-squared: 0.1752, Adjusted R-squared: 0.1652
F-statistic: 17.42 on 3 and 246 DF, p-value: 2.738e-10
On the previous slide, the interaction term was mean-centered. What if we use the raw age variable instead?
Call:
lm(formula = cognitive_flexibility ~ multilingual * age)
Residuals:
Min 1Q Median 3Q Max
-45.937 -9.753 1.457 12.291 38.200
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 29.321 10.350 2.833 0.0050 **
multilingual -16.799 14.351 -1.171 0.2429
age 2.316 1.079 2.145 0.0329 *
multilingual:age 2.949 1.506 1.959 0.0512 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 17.51 on 246 degrees of freedom
Multiple R-squared: 0.1752, Adjusted R-squared: 0.1652
F-statistic: 17.42 on 3 and 246 DF, p-value: 2.738e-10
marginaleffects: predictionslibrary(marginaleffects)
uncentered_model <- lm(cognitive_flexibility ~ multilingual * age, data = d)
preds <- predictions(uncentered_model)
preds |>
as.data.frame() |>
dplyr::select(rowid, multilingual, age, estimate, conf.low, conf.high) |>
knitr::kable(digits = 2)| rowid | multilingual | age | estimate | conf.low | conf.high |
|---|---|---|---|---|---|
| 1 | monolingual | 8.29 | 48.51 | 44.52 | 52.50 |
| 2 | monolingual | 8.11 | 48.10 | 43.86 | 52.34 |
| 3 | monolingual | 9.97 | 52.40 | 49.14 | 55.66 |
| 4 | multilingual | 8.34 | 56.42 | 52.73 | 60.12 |
| 5 | monolingual | 9.66 | 51.68 | 48.56 | 54.80 |
| 6 | multilingual | 10.93 | 70.05 | 65.61 | 74.49 |
| 7 | multilingual | 7.84 | 53.80 | 49.45 | 58.16 |
| 8 | multilingual | 9.02 | 60.02 | 56.90 | 63.15 |
| 9 | monolingual | 9.36 | 50.99 | 47.89 | 54.10 |
| 10 | monolingual | 11.34 | 55.58 | 50.57 | 60.59 |
| 11 | multilingual | 11.63 | 73.75 | 68.17 | 79.33 |
| 12 | multilingual | 11.41 | 72.60 | 67.39 | 77.81 |
| 13 | multilingual | 10.37 | 67.13 | 63.43 | 70.82 |
| 14 | monolingual | 11.75 | 56.53 | 50.81 | 62.25 |
| 15 | multilingual | 9.58 | 62.97 | 59.89 | 66.06 |
| 16 | monolingual | 9.88 | 52.21 | 49.00 | 55.42 |
| 17 | multilingual | 8.68 | 58.23 | 54.89 | 61.58 |
| 18 | monolingual | 8.74 | 49.55 | 46.08 | 53.02 |
| 19 | monolingual | 7.10 | 45.76 | 39.86 | 51.67 |
| 20 | monolingual | 9.51 | 51.35 | 48.26 | 54.45 |
| 21 | monolingual | 11.36 | 55.62 | 50.58 | 60.65 |
| 22 | multilingual | 7.03 | 49.54 | 43.88 | 55.21 |
| 23 | monolingual | 7.36 | 46.37 | 40.92 | 51.81 |
| 24 | monolingual | 7.82 | 47.43 | 42.76 | 52.11 |
| 25 | monolingual | 10.85 | 54.45 | 50.20 | 58.70 |
| 26 | monolingual | 10.68 | 54.04 | 50.04 | 58.04 |
| 27 | multilingual | 11.86 | 74.96 | 68.98 | 80.95 |
| 28 | multilingual | 9.33 | 61.66 | 58.61 | 64.70 |
| 29 | monolingual | 7.37 | 46.39 | 40.97 | 51.82 |
| 30 | multilingual | 10.24 | 66.46 | 62.91 | 70.01 |
| 31 | monolingual | 10.79 | 54.31 | 50.15 | 58.48 |
| 32 | multilingual | 7.69 | 52.99 | 48.40 | 57.58 |
| 33 | monolingual | 8.98 | 50.12 | 46.86 | 53.39 |
| 34 | multilingual | 8.12 | 55.30 | 51.34 | 59.26 |
| 35 | multilingual | 7.29 | 50.90 | 45.68 | 56.13 |
| 36 | monolingual | 8.98 | 50.12 | 46.85 | 53.38 |
| 37 | monolingual | 7.32 | 46.28 | 40.78 | 51.79 |
| 38 | monolingual | 8.13 | 48.15 | 43.94 | 52.35 |
| 39 | monolingual | 7.27 | 46.16 | 40.57 | 51.76 |
| 40 | multilingual | 10.35 | 67.02 | 63.35 | 70.70 |
| 41 | monolingual | 8.49 | 48.98 | 45.24 | 52.71 |
| 42 | multilingual | 7.50 | 52.03 | 47.15 | 56.91 |
| 43 | multilingual | 7.36 | 51.27 | 46.16 | 56.38 |
| 44 | monolingual | 11.40 | 55.73 | 50.61 | 60.84 |
| 45 | monolingual | 10.77 | 54.26 | 50.13 | 58.40 |
| 46 | monolingual | 11.08 | 54.99 | 50.39 | 59.58 |
| 47 | monolingual | 11.91 | 56.90 | 50.90 | 62.91 |
| 48 | multilingual | 7.52 | 52.11 | 47.25 | 56.96 |
| 49 | monolingual | 7.50 | 46.68 | 41.47 | 51.89 |
| 50 | monolingual | 10.99 | 54.78 | 50.32 | 59.24 |
| 51 | multilingual | 10.92 | 70.03 | 65.60 | 74.47 |
| 52 | monolingual | 7.05 | 45.64 | 39.64 | 51.64 |
| 53 | monolingual | 10.90 | 54.55 | 50.24 | 58.86 |
| 54 | monolingual | 10.65 | 53.98 | 50.01 | 57.94 |
| 55 | monolingual | 10.15 | 52.83 | 49.42 | 56.23 |
| 56 | multilingual | 9.40 | 62.04 | 58.99 | 65.09 |
| 57 | multilingual | 7.78 | 53.50 | 49.06 | 57.94 |
| 58 | monolingual | 7.04 | 45.63 | 39.62 | 51.64 |
| 59 | multilingual | 9.26 | 61.29 | 58.24 | 64.34 |
| 60 | monolingual | 9.46 | 51.23 | 48.14 | 54.33 |
| 61 | monolingual | 8.95 | 50.04 | 46.75 | 53.33 |
| 62 | multilingual | 9.32 | 61.61 | 58.57 | 64.66 |
| 63 | multilingual | 10.57 | 68.16 | 64.22 | 72.09 |
| 64 | monolingual | 7.28 | 46.17 | 40.58 | 51.76 |
| 65 | monolingual | 8.77 | 49.64 | 46.21 | 53.07 |
| 66 | multilingual | 11.01 | 70.51 | 65.94 | 75.09 |
| 67 | monolingual | 11.18 | 55.21 | 50.46 | 59.95 |
| 68 | monolingual | 8.19 | 48.28 | 44.16 | 52.41 |
| 69 | monolingual | 8.77 | 49.63 | 46.19 | 53.07 |
| 70 | monolingual | 11.28 | 55.45 | 50.53 | 60.37 |
| 71 | multilingual | 11.27 | 71.85 | 66.88 | 76.83 |
| 72 | monolingual | 8.48 | 48.96 | 45.21 | 52.70 |
| 73 | monolingual | 7.74 | 47.23 | 42.42 | 52.05 |
| 74 | monolingual | 10.52 | 53.68 | 49.88 | 57.48 |
| 75 | monolingual | 7.52 | 46.73 | 41.56 | 51.90 |
| 76 | multilingual | 7.17 | 50.27 | 44.84 | 55.70 |
| 77 | multilingual | 12.00 | 75.69 | 69.46 | 81.92 |
| 78 | monolingual | 7.17 | 45.94 | 40.16 | 51.71 |
| 79 | multilingual | 8.69 | 58.29 | 54.95 | 61.62 |
| 80 | multilingual | 11.58 | 73.47 | 67.98 | 78.96 |
| 81 | multilingual | 10.09 | 65.63 | 62.23 | 69.02 |
| 82 | multilingual | 8.43 | 56.92 | 53.33 | 60.50 |
| 83 | monolingual | 10.69 | 54.07 | 50.06 | 58.09 |
| 84 | multilingual | 11.17 | 71.34 | 66.52 | 76.16 |
| 85 | multilingual | 8.57 | 57.65 | 54.21 | 61.10 |
| 86 | multilingual | 9.46 | 62.35 | 59.29 | 65.40 |
| 87 | monolingual | 10.49 | 53.61 | 49.85 | 57.37 |
| 88 | monolingual | 10.21 | 52.96 | 49.50 | 56.42 |
| 89 | multilingual | 10.22 | 66.33 | 62.80 | 69.86 |
| 90 | monolingual | 11.89 | 56.85 | 50.89 | 62.82 |
| 91 | multilingual | 9.07 | 60.30 | 57.20 | 63.40 |
| 92 | multilingual | 7.60 | 52.52 | 47.79 | 57.25 |
| 93 | monolingual | 9.63 | 51.62 | 48.51 | 54.73 |
| 94 | multilingual | 8.13 | 55.30 | 51.35 | 59.26 |
| 95 | multilingual | 9.43 | 62.18 | 59.13 | 65.23 |
| 96 | monolingual | 8.85 | 49.82 | 46.45 | 53.18 |
| 97 | monolingual | 11.92 | 56.92 | 50.90 | 62.94 |
| 98 | multilingual | 8.94 | 59.60 | 56.44 | 62.76 |
| 99 | monolingual | 8.15 | 48.19 | 44.00 | 52.37 |
| 100 | monolingual | 10.12 | 52.75 | 49.37 | 56.13 |
| 101 | monolingual | 7.68 | 47.11 | 42.22 | 52.01 |
| 102 | multilingual | 11.84 | 74.85 | 68.90 | 80.80 |
| 103 | multilingual | 9.58 | 62.94 | 59.86 | 66.02 |
| 104 | monolingual | 7.82 | 47.42 | 42.74 | 52.10 |
| 105 | multilingual | 10.11 | 65.75 | 62.33 | 69.17 |
| 106 | multilingual | 11.93 | 75.33 | 69.22 | 81.45 |
| 107 | monolingual | 10.34 | 53.27 | 49.68 | 56.87 |
| 108 | monolingual | 9.09 | 50.38 | 47.18 | 53.58 |
| 109 | monolingual | 8.62 | 49.28 | 45.69 | 52.86 |
| 110 | multilingual | 11.18 | 71.37 | 66.54 | 76.20 |
| 111 | multilingual | 7.72 | 53.16 | 48.63 | 57.70 |
| 112 | multilingual | 7.96 | 54.45 | 50.28 | 58.63 |
| 113 | multilingual | 11.48 | 72.99 | 67.65 | 78.32 |
| 114 | monolingual | 8.54 | 49.10 | 45.42 | 52.77 |
| 115 | monolingual | 8.82 | 49.74 | 46.34 | 53.13 |
| 116 | multilingual | 10.92 | 70.02 | 65.58 | 74.45 |
| 117 | multilingual | 7.97 | 54.47 | 50.30 | 58.64 |
| 118 | multilingual | 7.09 | 49.85 | 44.28 | 55.41 |
| 119 | monolingual | 9.03 | 50.24 | 47.01 | 53.47 |
| 120 | multilingual | 9.42 | 62.10 | 59.05 | 65.15 |
| 121 | monolingual | 9.11 | 50.42 | 47.23 | 53.61 |
| 122 | multilingual | 8.71 | 58.40 | 55.09 | 61.72 |
| 123 | monolingual | 11.33 | 55.56 | 50.57 | 60.56 |
| 124 | monolingual | 9.28 | 50.80 | 47.68 | 53.92 |
| 125 | monolingual | 9.67 | 51.71 | 48.59 | 54.83 |
| 126 | multilingual | 11.82 | 74.75 | 68.84 | 80.67 |
| 127 | multilingual | 10.87 | 69.77 | 65.41 | 74.13 |
| 128 | monolingual | 8.04 | 47.95 | 43.62 | 52.28 |
| 129 | monolingual | 8.54 | 49.11 | 45.44 | 52.78 |
| 130 | monolingual | 11.86 | 56.78 | 50.87 | 62.69 |
| 131 | multilingual | 9.92 | 64.78 | 61.51 | 68.04 |
| 132 | monolingual | 10.80 | 54.34 | 50.16 | 58.52 |
| 133 | multilingual | 8.86 | 59.19 | 55.98 | 62.40 |
| 134 | multilingual | 10.85 | 69.63 | 65.31 | 73.95 |
| 135 | multilingual | 9.69 | 63.53 | 60.41 | 66.66 |
| 136 | multilingual | 11.57 | 73.44 | 67.96 | 78.92 |
| 137 | multilingual | 7.93 | 54.26 | 50.03 | 58.49 |
| 138 | multilingual | 8.41 | 56.81 | 53.20 | 60.42 |
| 139 | multilingual | 7.47 | 51.88 | 46.96 | 56.80 |
| 140 | monolingual | 8.05 | 47.97 | 43.65 | 52.29 |
| 141 | multilingual | 11.89 | 75.10 | 69.07 | 81.13 |
| 142 | multilingual | 8.48 | 57.18 | 53.64 | 60.71 |
| 143 | multilingual | 10.63 | 68.49 | 64.47 | 72.51 |
| 144 | monolingual | 10.93 | 54.63 | 50.27 | 58.99 |
| 145 | multilingual | 7.53 | 52.15 | 47.32 | 56.99 |
| 146 | multilingual | 8.20 | 55.69 | 51.82 | 59.55 |
| 147 | multilingual | 8.35 | 56.50 | 52.82 | 60.18 |
| 148 | monolingual | 7.51 | 46.70 | 41.51 | 51.89 |
| 149 | multilingual | 7.59 | 52.48 | 47.74 | 57.22 |
| 150 | multilingual | 11.96 | 75.47 | 69.32 | 81.63 |
| 151 | monolingual | 11.93 | 56.95 | 50.91 | 62.99 |
| 152 | multilingual | 7.69 | 52.99 | 48.40 | 57.58 |
| 153 | monolingual | 11.53 | 56.01 | 50.69 | 61.34 |
| 154 | multilingual | 9.88 | 64.55 | 61.32 | 67.78 |
| 155 | monolingual | 8.98 | 50.11 | 46.84 | 53.38 |
| 156 | monolingual | 9.25 | 50.74 | 47.61 | 53.87 |
| 157 | monolingual | 10.53 | 53.71 | 49.90 | 57.53 |
| 158 | monolingual | 7.41 | 46.49 | 41.13 | 51.84 |
| 159 | monolingual | 8.70 | 49.46 | 45.95 | 52.97 |
| 160 | monolingual | 10.40 | 53.41 | 49.75 | 57.08 |
| 161 | monolingual | 8.58 | 49.20 | 45.58 | 52.83 |
| 162 | multilingual | 11.16 | 71.27 | 66.47 | 76.07 |
| 163 | multilingual | 8.08 | 55.04 | 51.02 | 59.07 |
| 164 | multilingual | 9.49 | 62.49 | 59.43 | 65.55 |
| 165 | monolingual | 8.38 | 48.73 | 44.86 | 52.59 |
| 166 | multilingual | 7.96 | 54.43 | 50.25 | 58.62 |
| 167 | monolingual | 11.75 | 56.54 | 50.81 | 62.26 |
| 168 | monolingual | 8.61 | 49.26 | 45.66 | 52.85 |
| 169 | multilingual | 9.39 | 61.97 | 58.93 | 65.02 |
| 170 | monolingual | 7.14 | 45.86 | 40.02 | 51.69 |
| 171 | multilingual | 9.74 | 63.79 | 60.65 | 66.94 |
| 172 | multilingual | 10.22 | 66.34 | 62.81 | 69.87 |
| 173 | multilingual | 9.98 | 65.08 | 61.77 | 68.38 |
| 174 | multilingual | 8.61 | 57.85 | 54.44 | 61.26 |
| 175 | monolingual | 11.46 | 55.85 | 50.64 | 61.05 |
| 176 | monolingual | 10.13 | 52.78 | 49.39 | 56.17 |
| 177 | monolingual | 8.51 | 49.04 | 45.33 | 52.74 |
| 178 | multilingual | 8.94 | 59.60 | 56.44 | 62.76 |
| 179 | multilingual | 7.80 | 53.60 | 49.19 | 58.02 |
| 180 | multilingual | 11.31 | 72.09 | 67.04 | 77.14 |
| 181 | multilingual | 11.77 | 74.47 | 68.65 | 80.29 |
| 182 | monolingual | 9.82 | 52.06 | 48.88 | 55.24 |
| 183 | monolingual | 8.65 | 49.35 | 45.79 | 52.90 |
| 184 | monolingual | 11.98 | 57.07 | 50.93 | 63.21 |
| 185 | monolingual | 8.17 | 48.25 | 44.11 | 52.39 |
| 186 | multilingual | 10.06 | 65.51 | 62.13 | 68.88 |
| 187 | multilingual | 7.54 | 52.23 | 47.41 | 57.04 |
| 188 | monolingual | 9.44 | 51.17 | 48.07 | 54.27 |
| 189 | monolingual | 7.50 | 46.68 | 41.48 | 51.89 |
| 190 | multilingual | 7.81 | 53.62 | 49.21 | 58.03 |
| 191 | multilingual | 8.41 | 56.83 | 53.22 | 60.44 |
| 192 | monolingual | 9.92 | 52.29 | 49.06 | 55.52 |
| 193 | monolingual | 10.66 | 54.00 | 50.03 | 57.98 |
| 194 | multilingual | 7.83 | 53.74 | 49.36 | 58.11 |
| 195 | multilingual | 11.33 | 72.19 | 67.11 | 77.27 |
| 196 | monolingual | 10.54 | 53.74 | 49.91 | 57.56 |
| 197 | monolingual | 10.80 | 54.34 | 50.16 | 58.51 |
| 198 | multilingual | 7.74 | 53.25 | 48.74 | 57.76 |
| 199 | monolingual | 8.79 | 49.68 | 46.26 | 53.10 |
| 200 | multilingual | 10.37 | 67.10 | 63.42 | 70.79 |
| 201 | multilingual | 9.62 | 63.17 | 60.08 | 66.26 |
| 202 | multilingual | 8.75 | 58.59 | 55.30 | 61.88 |
| 203 | multilingual | 8.20 | 55.71 | 51.85 | 59.57 |
| 204 | monolingual | 7.29 | 46.21 | 40.64 | 51.77 |
| 205 | multilingual | 8.18 | 55.61 | 51.72 | 59.49 |
| 206 | multilingual | 11.45 | 72.81 | 67.53 | 78.09 |
| 207 | multilingual | 11.06 | 70.75 | 66.11 | 75.40 |
| 208 | monolingual | 10.74 | 54.19 | 50.10 | 58.27 |
| 209 | multilingual | 7.77 | 53.46 | 49.00 | 57.91 |
| 210 | multilingual | 7.62 | 52.66 | 47.98 | 57.35 |
| 211 | multilingual | 11.87 | 75.04 | 69.03 | 81.05 |
| 212 | monolingual | 9.18 | 50.58 | 47.42 | 53.74 |
| 213 | monolingual | 9.32 | 50.90 | 47.79 | 54.02 |
| 214 | multilingual | 7.83 | 53.73 | 49.35 | 58.11 |
| 215 | multilingual | 9.92 | 64.78 | 61.51 | 68.04 |
| 216 | multilingual | 8.35 | 56.51 | 52.83 | 60.18 |
| 217 | multilingual | 8.15 | 55.44 | 51.51 | 59.36 |
| 218 | monolingual | 10.46 | 53.53 | 49.81 | 57.26 |
| 219 | multilingual | 8.41 | 56.82 | 53.22 | 60.43 |
| 220 | monolingual | 11.05 | 54.91 | 50.37 | 59.46 |
| 221 | multilingual | 7.47 | 51.85 | 46.92 | 56.78 |
| 222 | monolingual | 11.11 | 55.05 | 50.41 | 59.69 |
| 223 | monolingual | 9.14 | 50.48 | 47.30 | 53.66 |
| 224 | monolingual | 10.78 | 54.28 | 50.14 | 58.43 |
| 225 | multilingual | 10.31 | 66.82 | 63.19 | 70.44 |
| 226 | multilingual | 9.22 | 61.08 | 58.02 | 64.14 |
| 227 | monolingual | 10.14 | 52.79 | 49.40 | 56.19 |
| 228 | monolingual | 7.00 | 45.54 | 39.46 | 51.62 |
| 229 | multilingual | 8.09 | 55.10 | 51.09 | 59.11 |
| 230 | monolingual | 10.52 | 53.69 | 49.89 | 57.50 |
| 231 | monolingual | 8.08 | 48.02 | 43.74 | 52.31 |
| 232 | multilingual | 11.07 | 70.81 | 66.14 | 75.47 |
| 233 | multilingual | 8.54 | 57.48 | 54.00 | 60.96 |
| 234 | monolingual | 10.44 | 53.49 | 49.79 | 57.20 |
| 235 | multilingual | 11.66 | 73.93 | 68.29 | 79.58 |
| 236 | multilingual | 7.58 | 52.43 | 47.67 | 57.18 |
| 237 | multilingual | 7.64 | 52.74 | 48.08 | 57.40 |
| 238 | multilingual | 10.39 | 67.23 | 63.52 | 70.95 |
| 239 | multilingual | 9.14 | 60.67 | 57.60 | 63.75 |
| 240 | monolingual | 11.17 | 55.19 | 50.46 | 59.93 |
| 241 | monolingual | 11.86 | 56.78 | 50.87 | 62.69 |
| 242 | multilingual | 7.35 | 51.23 | 46.11 | 56.36 |
| 243 | monolingual | 9.30 | 50.85 | 47.74 | 53.97 |
| 244 | multilingual | 10.51 | 67.85 | 63.99 | 71.71 |
| 245 | monolingual | 7.43 | 46.54 | 41.22 | 51.85 |
| 246 | monolingual | 11.96 | 57.03 | 50.92 | 63.13 |
| 247 | monolingual | 8.27 | 48.46 | 44.44 | 52.48 |
| 248 | multilingual | 7.25 | 50.68 | 45.39 | 55.98 |
| 249 | multilingual | 10.43 | 67.45 | 63.68 | 71.21 |
| 250 | multilingual | 10.93 | 70.10 | 65.64 | 74.55 |
marginaleffects: predictionslibrary(marginaleffects)
centered_model <- lm(cognitive_flexibility ~ multilingual * age_c, data = d)
preds <- predictions(centered_model)
preds |>
as.data.frame() |>
dplyr::select(rowid, multilingual, age_c, estimate, conf.low, conf.high) |>
knitr::kable(digits = 2)| rowid | multilingual | age_c | estimate | conf.low | conf.high |
|---|---|---|---|---|---|
| 1 | monolingual | -1.13 | 48.51 | 44.52 | 52.50 |
| 2 | monolingual | -1.31 | 48.10 | 43.86 | 52.34 |
| 3 | monolingual | 0.55 | 52.40 | 49.14 | 55.66 |
| 4 | multilingual | -1.08 | 56.42 | 52.73 | 60.12 |
| 5 | monolingual | 0.24 | 51.68 | 48.56 | 54.80 |
| 6 | multilingual | 1.51 | 70.05 | 65.61 | 74.49 |
| 7 | multilingual | -1.57 | 53.80 | 49.45 | 58.16 |
| 8 | multilingual | -0.39 | 60.02 | 56.90 | 63.15 |
| 9 | monolingual | -0.06 | 50.99 | 47.89 | 54.10 |
| 10 | monolingual | 1.93 | 55.58 | 50.57 | 60.59 |
| 11 | multilingual | 2.21 | 73.75 | 68.17 | 79.33 |
| 12 | multilingual | 2.00 | 72.60 | 67.39 | 77.81 |
| 13 | multilingual | 0.96 | 67.13 | 63.43 | 70.82 |
| 14 | monolingual | 2.34 | 56.53 | 50.81 | 62.25 |
| 15 | multilingual | 0.17 | 62.97 | 59.89 | 66.06 |
| 16 | monolingual | 0.47 | 52.21 | 49.00 | 55.42 |
| 17 | multilingual | -0.73 | 58.23 | 54.89 | 61.58 |
| 18 | monolingual | -0.68 | 49.55 | 46.08 | 53.02 |
| 19 | monolingual | -2.31 | 45.76 | 39.86 | 51.67 |
| 20 | monolingual | 0.10 | 51.35 | 48.26 | 54.45 |
| 21 | monolingual | 1.94 | 55.62 | 50.58 | 60.65 |
| 22 | multilingual | -2.38 | 49.54 | 43.88 | 55.21 |
| 23 | monolingual | -2.05 | 46.37 | 40.92 | 51.81 |
| 24 | monolingual | -1.59 | 47.43 | 42.76 | 52.11 |
| 25 | monolingual | 1.44 | 54.45 | 50.20 | 58.70 |
| 26 | monolingual | 1.26 | 54.04 | 50.04 | 58.04 |
| 27 | multilingual | 2.44 | 74.96 | 68.98 | 80.95 |
| 28 | multilingual | -0.08 | 61.66 | 58.61 | 64.70 |
| 29 | monolingual | -2.04 | 46.39 | 40.97 | 51.82 |
| 30 | multilingual | 0.83 | 66.46 | 62.91 | 70.01 |
| 31 | monolingual | 1.38 | 54.31 | 50.15 | 58.48 |
| 32 | multilingual | -1.73 | 52.99 | 48.40 | 57.58 |
| 33 | monolingual | -0.43 | 50.12 | 46.86 | 53.39 |
| 34 | multilingual | -1.29 | 55.30 | 51.34 | 59.26 |
| 35 | multilingual | -2.12 | 50.90 | 45.68 | 56.13 |
| 36 | monolingual | -0.44 | 50.12 | 46.85 | 53.38 |
| 37 | monolingual | -2.09 | 46.28 | 40.78 | 51.79 |
| 38 | monolingual | -1.29 | 48.15 | 43.94 | 52.35 |
| 39 | monolingual | -2.14 | 46.16 | 40.57 | 51.76 |
| 40 | multilingual | 0.94 | 67.02 | 63.35 | 70.70 |
| 41 | monolingual | -0.93 | 48.98 | 45.24 | 52.71 |
| 42 | multilingual | -1.91 | 52.03 | 47.15 | 56.91 |
| 43 | multilingual | -2.06 | 51.27 | 46.16 | 56.38 |
| 44 | monolingual | 1.99 | 55.73 | 50.61 | 60.84 |
| 45 | monolingual | 1.36 | 54.26 | 50.13 | 58.40 |
| 46 | monolingual | 1.67 | 54.99 | 50.39 | 59.58 |
| 47 | monolingual | 2.50 | 56.90 | 50.90 | 62.91 |
| 48 | multilingual | -1.90 | 52.11 | 47.25 | 56.96 |
| 49 | monolingual | -1.92 | 46.68 | 41.47 | 51.89 |
| 50 | monolingual | 1.58 | 54.78 | 50.32 | 59.24 |
| 51 | multilingual | 1.51 | 70.03 | 65.60 | 74.47 |
| 52 | monolingual | -2.37 | 45.64 | 39.64 | 51.64 |
| 53 | monolingual | 1.48 | 54.55 | 50.24 | 58.86 |
| 54 | monolingual | 1.23 | 53.98 | 50.01 | 57.94 |
| 55 | monolingual | 0.74 | 52.83 | 49.42 | 56.23 |
| 56 | multilingual | -0.01 | 62.04 | 58.99 | 65.09 |
| 57 | multilingual | -1.63 | 53.50 | 49.06 | 57.94 |
| 58 | monolingual | -2.37 | 45.63 | 39.62 | 51.64 |
| 59 | multilingual | -0.15 | 61.29 | 58.24 | 64.34 |
| 60 | monolingual | 0.05 | 51.23 | 48.14 | 54.33 |
| 61 | monolingual | -0.47 | 50.04 | 46.75 | 53.33 |
| 62 | multilingual | -0.09 | 61.61 | 58.57 | 64.66 |
| 63 | multilingual | 1.15 | 68.16 | 64.22 | 72.09 |
| 64 | monolingual | -2.14 | 46.17 | 40.58 | 51.76 |
| 65 | monolingual | -0.64 | 49.64 | 46.21 | 53.07 |
| 66 | multilingual | 1.60 | 70.51 | 65.94 | 75.09 |
| 67 | monolingual | 1.76 | 55.21 | 50.46 | 59.95 |
| 68 | monolingual | -1.23 | 48.28 | 44.16 | 52.41 |
| 69 | monolingual | -0.64 | 49.63 | 46.19 | 53.07 |
| 70 | monolingual | 1.87 | 55.45 | 50.53 | 60.37 |
| 71 | multilingual | 1.85 | 71.85 | 66.88 | 76.83 |
| 72 | monolingual | -0.94 | 48.96 | 45.21 | 52.70 |
| 73 | monolingual | -1.68 | 47.23 | 42.42 | 52.05 |
| 74 | monolingual | 1.11 | 53.68 | 49.88 | 57.48 |
| 75 | monolingual | -1.90 | 46.73 | 41.56 | 51.90 |
| 76 | multilingual | -2.25 | 50.27 | 44.84 | 55.70 |
| 77 | multilingual | 2.58 | 75.69 | 69.46 | 81.92 |
| 78 | monolingual | -2.24 | 45.94 | 40.16 | 51.71 |
| 79 | multilingual | -0.72 | 58.29 | 54.95 | 61.62 |
| 80 | multilingual | 2.16 | 73.47 | 67.98 | 78.96 |
| 81 | multilingual | 0.67 | 65.63 | 62.23 | 69.02 |
| 82 | multilingual | -0.98 | 56.92 | 53.33 | 60.50 |
| 83 | monolingual | 1.27 | 54.07 | 50.06 | 58.09 |
| 84 | multilingual | 1.76 | 71.34 | 66.52 | 76.16 |
| 85 | multilingual | -0.84 | 57.65 | 54.21 | 61.10 |
| 86 | multilingual | 0.05 | 62.35 | 59.29 | 65.40 |
| 87 | monolingual | 1.07 | 53.61 | 49.85 | 57.37 |
| 88 | monolingual | 0.79 | 52.96 | 49.50 | 56.42 |
| 89 | multilingual | 0.81 | 66.33 | 62.80 | 69.86 |
| 90 | monolingual | 2.47 | 56.85 | 50.89 | 62.82 |
| 91 | multilingual | -0.34 | 60.30 | 57.20 | 63.40 |
| 92 | multilingual | -1.82 | 52.52 | 47.79 | 57.25 |
| 93 | monolingual | 0.22 | 51.62 | 48.51 | 54.73 |
| 94 | multilingual | -1.29 | 55.30 | 51.35 | 59.26 |
| 95 | multilingual | 0.02 | 62.18 | 59.13 | 65.23 |
| 96 | monolingual | -0.56 | 49.82 | 46.45 | 53.18 |
| 97 | monolingual | 2.50 | 56.92 | 50.90 | 62.94 |
| 98 | multilingual | -0.47 | 59.60 | 56.44 | 62.76 |
| 99 | monolingual | -1.27 | 48.19 | 44.00 | 52.37 |
| 100 | monolingual | 0.70 | 52.75 | 49.37 | 56.13 |
| 101 | monolingual | -1.73 | 47.11 | 42.22 | 52.01 |
| 102 | multilingual | 2.42 | 74.85 | 68.90 | 80.80 |
| 103 | multilingual | 0.16 | 62.94 | 59.86 | 66.02 |
| 104 | monolingual | -1.60 | 47.42 | 42.74 | 52.10 |
| 105 | multilingual | 0.69 | 65.75 | 62.33 | 69.17 |
| 106 | multilingual | 2.52 | 75.33 | 69.22 | 81.45 |
| 107 | monolingual | 0.93 | 53.27 | 49.68 | 56.87 |
| 108 | monolingual | -0.32 | 50.38 | 47.18 | 53.58 |
| 109 | monolingual | -0.80 | 49.28 | 45.69 | 52.86 |
| 110 | multilingual | 1.76 | 71.37 | 66.54 | 76.20 |
| 111 | multilingual | -1.70 | 53.16 | 48.63 | 57.70 |
| 112 | multilingual | -1.45 | 54.45 | 50.28 | 58.63 |
| 113 | multilingual | 2.07 | 72.99 | 67.65 | 78.32 |
| 114 | monolingual | -0.87 | 49.10 | 45.42 | 52.77 |
| 115 | monolingual | -0.60 | 49.74 | 46.34 | 53.13 |
| 116 | multilingual | 1.51 | 70.02 | 65.58 | 74.45 |
| 117 | multilingual | -1.45 | 54.47 | 50.30 | 58.64 |
| 118 | multilingual | -2.33 | 49.85 | 44.28 | 55.41 |
| 119 | monolingual | -0.38 | 50.24 | 47.01 | 53.47 |
| 120 | multilingual | 0.00 | 62.10 | 59.05 | 65.15 |
| 121 | monolingual | -0.31 | 50.42 | 47.23 | 53.61 |
| 122 | multilingual | -0.70 | 58.40 | 55.09 | 61.72 |
| 123 | monolingual | 1.92 | 55.56 | 50.57 | 60.56 |
| 124 | monolingual | -0.14 | 50.80 | 47.68 | 53.92 |
| 125 | monolingual | 0.25 | 51.71 | 48.59 | 54.83 |
| 126 | multilingual | 2.40 | 74.75 | 68.84 | 80.67 |
| 127 | multilingual | 1.46 | 69.77 | 65.41 | 74.13 |
| 128 | monolingual | -1.37 | 47.95 | 43.62 | 52.28 |
| 129 | monolingual | -0.87 | 49.11 | 45.44 | 52.78 |
| 130 | monolingual | 2.44 | 56.78 | 50.87 | 62.69 |
| 131 | multilingual | 0.51 | 64.78 | 61.51 | 68.04 |
| 132 | monolingual | 1.39 | 54.34 | 50.16 | 58.52 |
| 133 | multilingual | -0.55 | 59.19 | 55.98 | 62.40 |
| 134 | multilingual | 1.43 | 69.63 | 65.31 | 73.95 |
| 135 | multilingual | 0.27 | 63.53 | 60.41 | 66.66 |
| 136 | multilingual | 2.16 | 73.44 | 67.96 | 78.92 |
| 137 | multilingual | -1.49 | 54.26 | 50.03 | 58.49 |
| 138 | multilingual | -1.00 | 56.81 | 53.20 | 60.42 |
| 139 | multilingual | -1.94 | 51.88 | 46.96 | 56.80 |
| 140 | monolingual | -1.36 | 47.97 | 43.65 | 52.29 |
| 141 | multilingual | 2.47 | 75.10 | 69.07 | 81.13 |
| 142 | multilingual | -0.93 | 57.18 | 53.64 | 60.71 |
| 143 | multilingual | 1.22 | 68.49 | 64.47 | 72.51 |
| 144 | monolingual | 1.51 | 54.63 | 50.27 | 58.99 |
| 145 | multilingual | -1.89 | 52.15 | 47.32 | 56.99 |
| 146 | multilingual | -1.22 | 55.69 | 51.82 | 59.55 |
| 147 | multilingual | -1.06 | 56.50 | 52.82 | 60.18 |
| 148 | monolingual | -1.91 | 46.70 | 41.51 | 51.89 |
| 149 | multilingual | -1.83 | 52.48 | 47.74 | 57.22 |
| 150 | multilingual | 2.54 | 75.47 | 69.32 | 81.63 |
| 151 | monolingual | 2.52 | 56.95 | 50.91 | 62.99 |
| 152 | multilingual | -1.73 | 52.99 | 48.40 | 57.58 |
| 153 | monolingual | 2.11 | 56.01 | 50.69 | 61.34 |
| 154 | multilingual | 0.47 | 64.55 | 61.32 | 67.78 |
| 155 | monolingual | -0.44 | 50.11 | 46.84 | 53.38 |
| 156 | monolingual | -0.17 | 50.74 | 47.61 | 53.87 |
| 157 | monolingual | 1.12 | 53.71 | 49.90 | 57.53 |
| 158 | monolingual | -2.00 | 46.49 | 41.13 | 51.84 |
| 159 | monolingual | -0.72 | 49.46 | 45.95 | 52.97 |
| 160 | monolingual | 0.99 | 53.41 | 49.75 | 57.08 |
| 161 | monolingual | -0.83 | 49.20 | 45.58 | 52.83 |
| 162 | multilingual | 1.74 | 71.27 | 66.47 | 76.07 |
| 163 | multilingual | -1.34 | 55.04 | 51.02 | 59.07 |
| 164 | multilingual | 0.08 | 62.49 | 59.43 | 65.55 |
| 165 | monolingual | -1.03 | 48.73 | 44.86 | 52.59 |
| 166 | multilingual | -1.45 | 54.43 | 50.25 | 58.62 |
| 167 | monolingual | 2.34 | 56.54 | 50.81 | 62.26 |
| 168 | monolingual | -0.81 | 49.26 | 45.66 | 52.85 |
| 169 | multilingual | -0.02 | 61.97 | 58.93 | 65.02 |
| 170 | monolingual | -2.27 | 45.86 | 40.02 | 51.69 |
| 171 | multilingual | 0.32 | 63.79 | 60.65 | 66.94 |
| 172 | multilingual | 0.81 | 66.34 | 62.81 | 69.87 |
| 173 | multilingual | 0.57 | 65.08 | 61.77 | 68.38 |
| 174 | multilingual | -0.80 | 57.85 | 54.44 | 61.26 |
| 175 | monolingual | 2.04 | 55.85 | 50.64 | 61.05 |
| 176 | monolingual | 0.72 | 52.78 | 49.39 | 56.17 |
| 177 | monolingual | -0.90 | 49.04 | 45.33 | 52.74 |
| 178 | multilingual | -0.47 | 59.60 | 56.44 | 62.76 |
| 179 | multilingual | -1.61 | 53.60 | 49.19 | 58.02 |
| 180 | multilingual | 1.90 | 72.09 | 67.04 | 77.14 |
| 181 | multilingual | 2.35 | 74.47 | 68.65 | 80.29 |
| 182 | monolingual | 0.40 | 52.06 | 48.88 | 55.24 |
| 183 | monolingual | -0.77 | 49.35 | 45.79 | 52.90 |
| 184 | monolingual | 2.57 | 57.07 | 50.93 | 63.21 |
| 185 | monolingual | -1.24 | 48.25 | 44.11 | 52.39 |
| 186 | multilingual | 0.65 | 65.51 | 62.13 | 68.88 |
| 187 | multilingual | -1.87 | 52.23 | 47.41 | 57.04 |
| 188 | monolingual | 0.02 | 51.17 | 48.07 | 54.27 |
| 189 | monolingual | -1.92 | 46.68 | 41.48 | 51.89 |
| 190 | multilingual | -1.61 | 53.62 | 49.21 | 58.03 |
| 191 | multilingual | -1.00 | 56.83 | 53.22 | 60.44 |
| 192 | monolingual | 0.50 | 52.29 | 49.06 | 55.52 |
| 193 | monolingual | 1.24 | 54.00 | 50.03 | 57.98 |
| 194 | multilingual | -1.59 | 53.74 | 49.36 | 58.11 |
| 195 | multilingual | 1.92 | 72.19 | 67.11 | 77.27 |
| 196 | monolingual | 1.13 | 53.74 | 49.91 | 57.56 |
| 197 | monolingual | 1.39 | 54.34 | 50.16 | 58.51 |
| 198 | multilingual | -1.68 | 53.25 | 48.74 | 57.76 |
| 199 | monolingual | -0.62 | 49.68 | 46.26 | 53.10 |
| 200 | multilingual | 0.95 | 67.10 | 63.42 | 70.79 |
| 201 | multilingual | 0.20 | 63.17 | 60.08 | 66.26 |
| 202 | multilingual | -0.67 | 58.59 | 55.30 | 61.88 |
| 203 | multilingual | -1.21 | 55.71 | 51.85 | 59.57 |
| 204 | monolingual | -2.12 | 46.21 | 40.64 | 51.77 |
| 205 | multilingual | -1.23 | 55.61 | 51.72 | 59.49 |
| 206 | multilingual | 2.04 | 72.81 | 67.53 | 78.09 |
| 207 | multilingual | 1.64 | 70.75 | 66.11 | 75.40 |
| 208 | monolingual | 1.32 | 54.19 | 50.10 | 58.27 |
| 209 | multilingual | -1.64 | 53.46 | 49.00 | 57.91 |
| 210 | multilingual | -1.79 | 52.66 | 47.98 | 57.35 |
| 211 | multilingual | 2.46 | 75.04 | 69.03 | 81.05 |
| 212 | monolingual | -0.23 | 50.58 | 47.42 | 53.74 |
| 213 | monolingual | -0.09 | 50.90 | 47.79 | 54.02 |
| 214 | multilingual | -1.59 | 53.73 | 49.35 | 58.11 |
| 215 | multilingual | 0.51 | 64.78 | 61.51 | 68.04 |
| 216 | multilingual | -1.06 | 56.51 | 52.83 | 60.18 |
| 217 | multilingual | -1.26 | 55.44 | 51.51 | 59.36 |
| 218 | monolingual | 1.04 | 53.53 | 49.81 | 57.26 |
| 219 | multilingual | -1.00 | 56.82 | 53.22 | 60.43 |
| 220 | monolingual | 1.64 | 54.91 | 50.37 | 59.46 |
| 221 | multilingual | -1.95 | 51.85 | 46.92 | 56.78 |
| 222 | monolingual | 1.70 | 55.05 | 50.41 | 59.69 |
| 223 | monolingual | -0.28 | 50.48 | 47.30 | 53.66 |
| 224 | monolingual | 1.36 | 54.28 | 50.14 | 58.43 |
| 225 | multilingual | 0.90 | 66.82 | 63.19 | 70.44 |
| 226 | multilingual | -0.19 | 61.08 | 58.02 | 64.14 |
| 227 | monolingual | 0.72 | 52.79 | 49.40 | 56.19 |
| 228 | monolingual | -2.41 | 45.54 | 39.46 | 51.62 |
| 229 | multilingual | -1.33 | 55.10 | 51.09 | 59.11 |
| 230 | monolingual | 1.11 | 53.69 | 49.89 | 57.50 |
| 231 | monolingual | -1.34 | 48.02 | 43.74 | 52.31 |
| 232 | multilingual | 1.66 | 70.81 | 66.14 | 75.47 |
| 233 | multilingual | -0.88 | 57.48 | 54.00 | 60.96 |
| 234 | monolingual | 1.02 | 53.49 | 49.79 | 57.20 |
| 235 | multilingual | 2.25 | 73.93 | 68.29 | 79.58 |
| 236 | multilingual | -1.84 | 52.43 | 47.67 | 57.18 |
| 237 | multilingual | -1.78 | 52.74 | 48.08 | 57.40 |
| 238 | multilingual | 0.98 | 67.23 | 63.52 | 70.95 |
| 239 | multilingual | -0.27 | 60.67 | 57.60 | 63.75 |
| 240 | monolingual | 1.76 | 55.19 | 50.46 | 59.93 |
| 241 | monolingual | 2.44 | 56.78 | 50.87 | 62.69 |
| 242 | multilingual | -2.06 | 51.23 | 46.11 | 56.36 |
| 243 | monolingual | -0.12 | 50.85 | 47.74 | 53.97 |
| 244 | multilingual | 1.09 | 67.85 | 63.99 | 71.71 |
| 245 | monolingual | -1.98 | 46.54 | 41.22 | 51.85 |
| 246 | monolingual | 2.55 | 57.03 | 50.92 | 63.13 |
| 247 | monolingual | -1.15 | 48.46 | 44.44 | 52.48 |
| 248 | multilingual | -2.17 | 50.68 | 45.39 | 55.98 |
| 249 | multilingual | 1.02 | 67.45 | 63.68 | 71.21 |
| 250 | multilingual | 1.52 | 70.10 | 65.64 | 74.55 |
comparisons <- comparisons(uncentered_model, variables = "multilingual")
comparisons |>
as.data.frame() |>
dplyr::select(rowid, contrast, age, estimate, conf.low, conf.high) |>
knitr::kable(digits = 2)| rowid | contrast | age | estimate | conf.low | conf.high |
|---|---|---|---|---|---|
| 1 | multilingual - monolingual | 8.29 | 7.64 | 2.16 | 13.12 |
| 2 | multilingual - monolingual | 8.11 | 7.12 | 1.31 | 12.93 |
| 3 | multilingual - monolingual | 9.97 | 12.59 | 7.96 | 17.23 |
| 4 | multilingual - monolingual | 8.34 | 7.79 | 2.40 | 13.18 |
| 5 | multilingual - monolingual | 9.66 | 11.68 | 7.28 | 16.08 |
| 6 | multilingual - monolingual | 10.93 | 15.43 | 9.21 | 21.65 |
| 7 | multilingual - monolingual | 7.84 | 6.33 | -0.04 | 12.69 |
| 8 | multilingual - monolingual | 9.02 | 9.81 | 5.31 | 14.31 |
| 9 | multilingual - monolingual | 9.36 | 10.80 | 6.45 | 15.15 |
| 10 | multilingual - monolingual | 11.34 | 16.65 | 9.50 | 23.80 |
| 11 | multilingual - monolingual | 11.63 | 17.50 | 9.66 | 25.34 |
| 12 | multilingual - monolingual | 11.41 | 16.85 | 9.54 | 24.17 |
| 13 | multilingual - monolingual | 10.37 | 13.79 | 8.61 | 18.97 |
| 14 | multilingual - monolingual | 11.75 | 17.86 | 9.72 | 26.00 |
| 15 | multilingual - monolingual | 9.58 | 11.46 | 7.09 | 15.84 |
| 16 | multilingual - monolingual | 9.88 | 12.35 | 7.79 | 16.91 |
| 17 | multilingual - monolingual | 8.68 | 8.81 | 3.95 | 13.66 |
| 18 | multilingual - monolingual | 8.74 | 8.97 | 4.18 | 13.76 |
| 19 | multilingual - monolingual | 7.10 | 4.14 | -3.96 | 12.24 |
| 20 | multilingual - monolingual | 9.51 | 11.26 | 6.91 | 15.62 |
| 21 | multilingual - monolingual | 11.36 | 16.69 | 9.51 | 23.88 |
| 22 | multilingual - monolingual | 7.03 | 3.94 | -4.33 | 12.21 |
| 23 | multilingual - monolingual | 7.36 | 4.91 | -2.56 | 12.37 |
| 24 | multilingual - monolingual | 7.82 | 6.27 | -0.14 | 12.68 |
| 25 | multilingual - monolingual | 10.85 | 15.21 | 9.14 | 21.27 |
| 26 | multilingual - monolingual | 10.68 | 14.69 | 8.97 | 20.41 |
| 27 | multilingual - monolingual | 11.86 | 18.18 | 9.76 | 26.59 |
| 28 | multilingual - monolingual | 9.33 | 10.73 | 6.37 | 15.08 |
| 29 | multilingual - monolingual | 7.37 | 4.94 | -2.49 | 12.38 |
| 30 | multilingual - monolingual | 10.24 | 13.42 | 8.43 | 18.40 |
| 31 | multilingual - monolingual | 10.79 | 15.03 | 9.09 | 20.98 |
| 32 | multilingual - monolingual | 7.69 | 5.87 | -0.84 | 12.58 |
| 33 | multilingual - monolingual | 8.98 | 9.70 | 5.17 | 14.23 |
| 34 | multilingual - monolingual | 8.12 | 7.17 | 1.38 | 12.95 |
| 35 | multilingual - monolingual | 7.29 | 4.70 | -2.93 | 12.34 |
| 36 | multilingual - monolingual | 8.98 | 9.69 | 5.15 | 14.22 |
| 37 | multilingual - monolingual | 7.32 | 4.80 | -2.75 | 12.36 |
| 38 | multilingual - monolingual | 8.13 | 7.18 | 1.41 | 12.95 |
| 39 | multilingual - monolingual | 7.27 | 4.65 | -3.02 | 12.33 |
| 40 | multilingual - monolingual | 10.35 | 13.73 | 8.59 | 18.88 |
| 41 | multilingual - monolingual | 8.49 | 8.24 | 3.10 | 13.38 |
| 42 | multilingual - monolingual | 7.50 | 5.33 | -1.79 | 12.46 |
| 43 | multilingual - monolingual | 7.36 | 4.91 | -2.56 | 12.37 |
| 44 | multilingual - monolingual | 11.40 | 16.83 | 9.54 | 24.12 |
| 45 | multilingual - monolingual | 10.77 | 14.97 | 9.07 | 20.87 |
| 46 | multilingual - monolingual | 11.08 | 15.89 | 9.33 | 22.45 |
| 47 | multilingual - monolingual | 11.91 | 18.33 | 9.79 | 26.88 |
| 48 | multilingual - monolingual | 7.52 | 5.37 | -1.72 | 12.47 |
| 49 | multilingual - monolingual | 7.50 | 5.31 | -1.84 | 12.45 |
| 50 | multilingual - monolingual | 10.99 | 15.63 | 9.26 | 21.99 |
| 51 | multilingual - monolingual | 10.92 | 15.42 | 9.20 | 21.63 |
| 52 | multilingual - monolingual | 7.05 | 3.99 | -4.25 | 12.22 |
| 53 | multilingual - monolingual | 10.90 | 15.34 | 9.18 | 21.49 |
| 54 | multilingual - monolingual | 10.65 | 14.60 | 8.94 | 20.26 |
| 55 | multilingual - monolingual | 10.15 | 13.14 | 8.29 | 17.99 |
| 56 | multilingual - monolingual | 9.40 | 10.94 | 6.59 | 15.28 |
| 57 | multilingual - monolingual | 7.78 | 6.16 | -0.33 | 12.65 |
| 58 | multilingual - monolingual | 7.04 | 3.97 | -4.28 | 12.22 |
| 59 | multilingual - monolingual | 9.26 | 10.52 | 6.15 | 14.89 |
| 60 | multilingual - monolingual | 9.46 | 11.11 | 6.76 | 15.45 |
| 61 | multilingual - monolingual | 8.95 | 9.59 | 5.03 | 14.15 |
| 62 | multilingual - monolingual | 9.32 | 10.70 | 6.35 | 15.05 |
| 63 | multilingual - monolingual | 10.57 | 14.37 | 8.85 | 19.88 |
| 64 | multilingual - monolingual | 7.28 | 4.66 | -3.00 | 12.33 |
| 65 | multilingual - monolingual | 8.77 | 9.08 | 4.34 | 13.82 |
| 66 | multilingual - monolingual | 11.01 | 15.69 | 9.28 | 22.10 |
| 67 | multilingual - monolingual | 11.18 | 16.17 | 9.40 | 22.95 |
| 68 | multilingual - monolingual | 8.19 | 7.35 | 1.69 | 13.01 |
| 69 | multilingual - monolingual | 8.77 | 9.07 | 4.32 | 13.81 |
| 70 | multilingual - monolingual | 11.28 | 16.48 | 9.47 | 23.50 |
| 71 | multilingual - monolingual | 11.27 | 16.44 | 9.46 | 23.42 |
| 72 | multilingual - monolingual | 8.48 | 8.21 | 3.06 | 13.36 |
| 73 | multilingual - monolingual | 7.74 | 6.02 | -0.58 | 12.61 |
| 74 | multilingual - monolingual | 10.52 | 14.23 | 8.80 | 19.66 |
| 75 | multilingual - monolingual | 7.52 | 5.38 | -1.71 | 12.47 |
| 76 | multilingual - monolingual | 7.17 | 4.34 | -3.59 | 12.28 |
| 77 | multilingual - monolingual | 12.00 | 18.59 | 9.82 | 27.35 |
| 78 | multilingual - monolingual | 7.17 | 4.36 | -3.56 | 12.28 |
| 79 | multilingual - monolingual | 8.69 | 8.84 | 3.99 | 13.68 |
| 80 | multilingual - monolingual | 11.58 | 17.34 | 9.63 | 25.05 |
| 81 | multilingual - monolingual | 10.09 | 12.95 | 8.18 | 17.72 |
| 82 | multilingual - monolingual | 8.43 | 8.07 | 2.84 | 13.30 |
| 83 | multilingual - monolingual | 10.69 | 14.73 | 8.99 | 20.47 |
| 84 | multilingual - monolingual | 11.17 | 16.15 | 9.39 | 22.90 |
| 85 | multilingual - monolingual | 8.57 | 8.48 | 3.47 | 13.49 |
| 86 | multilingual - monolingual | 9.46 | 11.11 | 6.76 | 15.46 |
| 87 | multilingual - monolingual | 10.49 | 14.13 | 8.76 | 19.50 |
| 88 | multilingual - monolingual | 10.21 | 13.31 | 8.38 | 18.24 |
| 89 | multilingual - monolingual | 10.22 | 13.34 | 8.40 | 18.29 |
| 90 | multilingual - monolingual | 11.89 | 18.27 | 9.78 | 26.76 |
| 91 | multilingual - monolingual | 9.07 | 9.96 | 5.50 | 14.43 |
| 92 | multilingual - monolingual | 7.60 | 5.61 | -1.30 | 12.52 |
| 93 | multilingual - monolingual | 9.63 | 11.60 | 7.21 | 16.00 |
| 94 | multilingual - monolingual | 8.13 | 7.17 | 1.39 | 12.95 |
| 95 | multilingual - monolingual | 9.43 | 11.02 | 6.68 | 15.37 |
| 96 | multilingual - monolingual | 8.85 | 9.31 | 4.65 | 13.96 |
| 97 | multilingual - monolingual | 11.92 | 18.35 | 9.79 | 26.91 |
| 98 | multilingual - monolingual | 8.94 | 9.57 | 5.01 | 14.14 |
| 99 | multilingual - monolingual | 8.15 | 7.23 | 1.49 | 12.97 |
| 100 | multilingual - monolingual | 10.12 | 13.04 | 8.23 | 17.85 |
| 101 | multilingual - monolingual | 7.68 | 5.86 | -0.85 | 12.58 |
| 102 | multilingual - monolingual | 11.84 | 18.11 | 9.76 | 26.47 |
| 103 | multilingual - monolingual | 9.58 | 11.44 | 7.07 | 15.81 |
| 104 | multilingual - monolingual | 7.82 | 6.25 | -0.17 | 12.67 |
| 105 | multilingual - monolingual | 10.11 | 13.02 | 8.22 | 17.82 |
| 106 | multilingual - monolingual | 11.93 | 18.39 | 9.79 | 26.98 |
| 107 | multilingual - monolingual | 10.34 | 13.71 | 8.58 | 18.84 |
| 108 | multilingual - monolingual | 9.09 | 10.03 | 5.58 | 14.47 |
| 109 | multilingual - monolingual | 8.62 | 8.62 | 3.67 | 13.56 |
| 110 | multilingual - monolingual | 11.18 | 16.16 | 9.40 | 22.93 |
| 111 | multilingual - monolingual | 7.72 | 5.97 | -0.66 | 12.60 |
| 112 | multilingual - monolingual | 7.96 | 6.69 | 0.59 | 12.80 |
| 113 | multilingual - monolingual | 11.48 | 17.07 | 9.58 | 24.56 |
| 114 | multilingual - monolingual | 8.54 | 8.39 | 3.33 | 13.45 |
| 115 | multilingual - monolingual | 8.82 | 9.20 | 4.51 | 13.90 |
| 116 | multilingual - monolingual | 10.92 | 15.41 | 9.20 | 21.62 |
| 117 | multilingual - monolingual | 7.97 | 6.70 | 0.60 | 12.80 |
| 118 | multilingual - monolingual | 7.09 | 4.11 | -4.02 | 12.24 |
| 119 | multilingual - monolingual | 9.03 | 9.84 | 5.35 | 14.33 |
| 120 | multilingual - monolingual | 9.42 | 10.97 | 6.63 | 15.32 |
| 121 | multilingual - monolingual | 9.11 | 10.07 | 5.63 | 14.51 |
| 122 | multilingual - monolingual | 8.71 | 8.90 | 4.09 | 13.72 |
| 123 | multilingual - monolingual | 11.33 | 16.63 | 9.50 | 23.75 |
| 124 | multilingual - monolingual | 9.28 | 10.56 | 6.19 | 14.92 |
| 125 | multilingual - monolingual | 9.67 | 11.72 | 7.31 | 16.13 |
| 126 | multilingual - monolingual | 11.82 | 18.06 | 9.75 | 26.38 |
| 127 | multilingual - monolingual | 10.87 | 15.27 | 9.16 | 21.38 |
| 128 | multilingual - monolingual | 8.04 | 6.93 | 0.99 | 12.87 |
| 129 | multilingual - monolingual | 8.54 | 8.40 | 3.35 | 13.45 |
| 130 | multilingual - monolingual | 11.86 | 18.17 | 9.76 | 26.58 |
| 131 | multilingual - monolingual | 9.92 | 12.47 | 7.88 | 17.07 |
| 132 | multilingual - monolingual | 10.80 | 15.07 | 9.10 | 21.04 |
| 133 | multilingual - monolingual | 8.86 | 9.34 | 4.70 | 13.99 |
| 134 | multilingual - monolingual | 10.85 | 15.19 | 9.14 | 21.25 |
| 135 | multilingual - monolingual | 9.69 | 11.78 | 7.36 | 16.19 |
| 136 | multilingual - monolingual | 11.57 | 17.33 | 9.63 | 25.02 |
| 137 | multilingual - monolingual | 7.93 | 6.58 | 0.40 | 12.76 |
| 138 | multilingual - monolingual | 8.41 | 8.01 | 2.75 | 13.27 |
| 139 | multilingual - monolingual | 7.47 | 5.25 | -1.95 | 12.44 |
| 140 | multilingual - monolingual | 8.05 | 6.95 | 1.03 | 12.88 |
| 141 | multilingual - monolingual | 11.89 | 18.26 | 9.78 | 26.74 |
| 142 | multilingual - monolingual | 8.48 | 8.22 | 3.07 | 13.37 |
| 143 | multilingual - monolingual | 10.63 | 14.55 | 8.92 | 20.18 |
| 144 | multilingual - monolingual | 10.93 | 15.43 | 9.21 | 21.66 |
| 145 | multilingual - monolingual | 7.53 | 5.40 | -1.67 | 12.47 |
| 146 | multilingual - monolingual | 8.20 | 7.38 | 1.74 | 13.02 |
| 147 | multilingual - monolingual | 8.35 | 7.84 | 2.48 | 13.20 |
| 148 | multilingual - monolingual | 7.51 | 5.34 | -1.78 | 12.46 |
| 149 | multilingual - monolingual | 7.59 | 5.59 | -1.34 | 12.51 |
| 150 | multilingual - monolingual | 11.96 | 18.47 | 9.80 | 27.13 |
| 151 | multilingual - monolingual | 11.93 | 18.39 | 9.79 | 26.98 |
| 152 | multilingual - monolingual | 7.69 | 5.87 | -0.84 | 12.58 |
| 153 | multilingual - monolingual | 11.53 | 17.20 | 9.61 | 24.79 |
| 154 | multilingual - monolingual | 9.88 | 12.35 | 7.79 | 16.90 |
| 155 | multilingual - monolingual | 8.98 | 9.68 | 5.14 | 14.21 |
| 156 | multilingual - monolingual | 9.25 | 10.48 | 6.11 | 14.85 |
| 157 | multilingual - monolingual | 10.53 | 14.27 | 8.82 | 19.72 |
| 158 | multilingual - monolingual | 7.41 | 5.06 | -2.28 | 12.40 |
| 159 | multilingual - monolingual | 8.70 | 8.85 | 4.01 | 13.69 |
| 160 | multilingual - monolingual | 10.40 | 13.89 | 8.66 | 19.12 |
| 161 | multilingual - monolingual | 8.58 | 8.52 | 3.53 | 13.51 |
| 162 | multilingual - monolingual | 11.16 | 16.11 | 9.38 | 22.84 |
| 163 | multilingual - monolingual | 8.08 | 7.02 | 1.14 | 12.90 |
| 164 | multilingual - monolingual | 9.49 | 11.19 | 6.84 | 15.54 |
| 165 | multilingual - monolingual | 8.38 | 7.92 | 2.60 | 13.23 |
| 166 | multilingual - monolingual | 7.96 | 6.68 | 0.57 | 12.79 |
| 167 | multilingual - monolingual | 11.75 | 17.87 | 9.72 | 26.01 |
| 168 | multilingual - monolingual | 8.61 | 8.59 | 3.63 | 13.55 |
| 169 | multilingual - monolingual | 9.39 | 10.90 | 6.56 | 15.25 |
| 170 | multilingual - monolingual | 7.14 | 4.26 | -3.74 | 12.26 |
| 171 | multilingual - monolingual | 9.74 | 11.92 | 7.47 | 16.37 |
| 172 | multilingual - monolingual | 10.22 | 13.35 | 8.40 | 18.30 |
| 173 | multilingual - monolingual | 9.98 | 12.64 | 7.99 | 17.29 |
| 174 | multilingual - monolingual | 8.61 | 8.59 | 3.64 | 13.55 |
| 175 | multilingual - monolingual | 11.46 | 16.99 | 9.57 | 24.41 |
| 176 | multilingual - monolingual | 10.13 | 13.08 | 8.25 | 17.91 |
| 177 | multilingual - monolingual | 8.51 | 8.31 | 3.22 | 13.41 |
| 178 | multilingual - monolingual | 8.94 | 9.57 | 5.01 | 14.14 |
| 179 | multilingual - monolingual | 7.80 | 6.21 | -0.24 | 12.66 |
| 180 | multilingual - monolingual | 11.31 | 16.57 | 9.48 | 23.65 |
| 181 | multilingual - monolingual | 11.77 | 17.90 | 9.72 | 26.08 |
| 182 | multilingual - monolingual | 9.82 | 12.16 | 7.66 | 16.66 |
| 183 | multilingual - monolingual | 8.65 | 8.71 | 3.80 | 13.61 |
| 184 | multilingual - monolingual | 11.98 | 18.54 | 9.81 | 27.27 |
| 185 | multilingual - monolingual | 8.17 | 7.31 | 1.62 | 13.00 |
| 186 | multilingual - monolingual | 10.06 | 12.88 | 8.14 | 17.63 |
| 187 | multilingual - monolingual | 7.54 | 5.44 | -1.60 | 12.48 |
| 188 | multilingual - monolingual | 9.44 | 11.03 | 6.68 | 15.38 |
| 189 | multilingual - monolingual | 7.50 | 5.31 | -1.83 | 12.45 |
| 190 | multilingual - monolingual | 7.81 | 6.22 | -0.22 | 12.67 |
| 191 | multilingual - monolingual | 8.41 | 8.02 | 2.76 | 13.28 |
| 192 | multilingual - monolingual | 9.92 | 12.46 | 7.87 | 17.05 |
| 193 | multilingual - monolingual | 10.66 | 14.64 | 8.95 | 20.32 |
| 194 | multilingual - monolingual | 7.83 | 6.29 | -0.11 | 12.68 |
| 195 | multilingual - monolingual | 11.33 | 16.63 | 9.50 | 23.75 |
| 196 | multilingual - monolingual | 10.54 | 14.30 | 8.83 | 19.77 |
| 197 | multilingual - monolingual | 10.80 | 15.06 | 9.10 | 21.03 |
| 198 | multilingual - monolingual | 7.74 | 6.02 | -0.58 | 12.61 |
| 199 | multilingual - monolingual | 8.79 | 9.13 | 4.40 | 13.85 |
| 200 | multilingual - monolingual | 10.37 | 13.78 | 8.61 | 18.95 |
| 201 | multilingual - monolingual | 9.62 | 11.57 | 7.19 | 15.96 |
| 202 | multilingual - monolingual | 8.75 | 9.01 | 4.23 | 13.78 |
| 203 | multilingual - monolingual | 8.20 | 7.39 | 1.76 | 13.03 |
| 204 | multilingual - monolingual | 7.29 | 4.71 | -2.93 | 12.34 |
| 205 | multilingual - monolingual | 8.18 | 7.34 | 1.67 | 13.01 |
| 206 | multilingual - monolingual | 11.45 | 16.97 | 9.57 | 24.38 |
| 207 | multilingual - monolingual | 11.06 | 15.82 | 9.31 | 22.33 |
| 208 | multilingual - monolingual | 10.74 | 14.87 | 9.03 | 20.71 |
| 209 | multilingual - monolingual | 7.77 | 6.13 | -0.38 | 12.64 |
| 210 | multilingual - monolingual | 7.62 | 5.69 | -1.16 | 12.53 |
| 211 | multilingual - monolingual | 11.87 | 18.22 | 9.77 | 26.67 |
| 212 | multilingual - monolingual | 9.18 | 10.28 | 5.88 | 14.68 |
| 213 | multilingual - monolingual | 9.32 | 10.69 | 6.34 | 15.04 |
| 214 | multilingual - monolingual | 7.83 | 6.28 | -0.11 | 12.68 |
| 215 | multilingual - monolingual | 9.92 | 12.47 | 7.88 | 17.07 |
| 216 | multilingual - monolingual | 8.35 | 7.84 | 2.48 | 13.20 |
| 217 | multilingual - monolingual | 8.15 | 7.24 | 1.51 | 12.97 |
| 218 | multilingual - monolingual | 10.46 | 14.04 | 8.72 | 19.36 |
| 219 | multilingual - monolingual | 8.41 | 8.02 | 2.76 | 13.28 |
| 220 | multilingual - monolingual | 11.05 | 15.80 | 9.31 | 22.29 |
| 221 | multilingual - monolingual | 7.47 | 5.23 | -1.97 | 12.44 |
| 222 | multilingual - monolingual | 11.11 | 15.97 | 9.35 | 22.59 |
| 223 | multilingual - monolingual | 9.14 | 10.15 | 5.73 | 14.57 |
| 224 | multilingual - monolingual | 10.78 | 14.99 | 9.08 | 20.91 |
| 225 | multilingual - monolingual | 10.31 | 13.62 | 8.53 | 18.70 |
| 226 | multilingual - monolingual | 9.22 | 10.40 | 6.02 | 14.79 |
| 227 | multilingual - monolingual | 10.14 | 13.10 | 8.26 | 17.93 |
| 228 | multilingual - monolingual | 7.00 | 3.85 | -4.49 | 12.20 |
| 229 | multilingual - monolingual | 8.09 | 7.05 | 1.19 | 12.91 |
| 230 | multilingual - monolingual | 10.52 | 14.24 | 8.81 | 19.68 |
| 231 | multilingual - monolingual | 8.08 | 7.02 | 1.14 | 12.90 |
| 232 | multilingual - monolingual | 11.07 | 15.85 | 9.32 | 22.38 |
| 233 | multilingual - monolingual | 8.54 | 8.39 | 3.33 | 13.45 |
| 234 | multilingual - monolingual | 10.44 | 13.99 | 8.70 | 19.28 |
| 235 | multilingual - monolingual | 11.66 | 17.60 | 9.68 | 25.53 |
| 236 | multilingual - monolingual | 7.58 | 5.55 | -1.40 | 12.51 |
| 237 | multilingual - monolingual | 7.64 | 5.73 | -1.08 | 12.54 |
| 238 | multilingual - monolingual | 10.39 | 13.85 | 8.64 | 19.06 |
| 239 | multilingual - monolingual | 9.14 | 10.17 | 5.75 | 14.59 |
| 240 | multilingual - monolingual | 11.17 | 16.15 | 9.39 | 22.91 |
| 241 | multilingual - monolingual | 11.86 | 18.17 | 9.76 | 26.58 |
| 242 | multilingual - monolingual | 7.35 | 4.89 | -2.60 | 12.37 |
| 243 | multilingual - monolingual | 9.30 | 10.63 | 6.27 | 14.99 |
| 244 | multilingual - monolingual | 10.51 | 14.19 | 8.79 | 19.60 |
| 245 | multilingual - monolingual | 7.43 | 5.13 | -2.16 | 12.42 |
| 246 | multilingual - monolingual | 11.96 | 18.49 | 9.81 | 27.17 |
| 247 | multilingual - monolingual | 8.27 | 7.58 | 2.06 | 13.10 |
| 248 | multilingual - monolingual | 7.25 | 4.58 | -3.16 | 12.32 |
| 249 | multilingual - monolingual | 10.43 | 13.97 | 8.69 | 19.25 |
| 250 | multilingual - monolingual | 10.93 | 15.45 | 9.21 | 21.69 |
Because the plots are made with ggplot2, we can customize them as we like.
plot_predictions(uncentered_model, condition = list("age", "multilingual")) +
theme(legend.title = element_blank()) +
scale_color_manual(values = c("monolingual" = "royalblue",
"multilingual" = "darkorange")) +
scale_fill_manual(values = c("monolingual" = "royalblue",
"multilingual" = "darkorange")) +
labs(x = "Age", y = "Cognitive flexibility score")GLMs such as logistic regression and Poisson regression are also supported by marginaleffects.
Good news for us, because everything interacts in a GLM due to the link function (e.g., logit, log, softmax).
\[ \text{Agent} \sim \text{Bernoulli}(\pi) \] \[ \pi = \text{logit}^{-1}(\alpha + \beta \times \text{experimental}) \]
Call:
glm(formula = agent ~ condition, family = binomial(link = "logit"),
data = d)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.5465 0.3789 1.443 0.1491
conditionexperimental 1.3253 0.6573 2.016 0.0438 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 67.48 on 59 degrees of freedom
Residual deviance: 62.99 on 58 degrees of freedom
AIC: 66.99
Number of Fisher Scoring iterations: 4
Even with just a single predictor, we cannot get our estimate from looking at model coefficients.
Due to the link function, the effect of the experimental condition depends on the intercept.
‘Mechanistic’ rather than conceptual interaction.
The marginal effect on the probability scale for a log odds ratio of 1.33 depends strongly on the intercept!
AME <- avg_comparisons(logistic_model, variables = "condition")
AME |>
as.data.frame() |>
dplyr::select(term, contrast, estimate, conf.low, conf.high) |>
print(digits = 2) term contrast estimate conf.low conf.high
1 condition mean(experimental) - mean(control) 0.23 0.022 0.44
For frequentist models, marginaleffects by default uses the delta method (Taylor series expansion)
For Bayesian models (e.g., fit with brms), marginaleffects, uses full posterior draws
More info: https://marginaleffects.com/chapters/uncertainty.html
Reporting model results using marginaleffects:
Example: relationship between population density and structural diversity in language (measured by normalized entropy)
Family: gaussian
Link function: identity
Formula:
structural_diversity ~ s(log_density, k = 8)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.676467 0.009797 69.05 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Approximate significance of smooth terms:
edf Ref.df F p-value
s(log_density) 6.221 6.798 49.08 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
R-sq.(adj) = 0.623 Deviance explained = 63.5%
GCV = 0.019914 Scale est. = 0.019195 n = 200
slopes_df <- slopes(gam_model) |>
as.data.frame() |>
dplyr::select(estimate, conf.low, conf.high, log_density, structural_diversity)
print(head(slopes_df), digits = 2) estimate conf.low conf.high log_density structural_diversity
1 0.1590 0.110 0.208 0.88 0.45
2 0.0023 -0.059 0.064 5.88 0.89
3 -0.0065 -0.059 0.046 2.09 0.68
4 0.0392 -0.020 0.098 6.83 0.83
5 0.0912 -0.015 0.197 7.40 0.96
6 0.0240 -0.100 0.148 -1.54 0.37
\(\mathbb{E}\left[\frac{\partial \text{structural diversity}}{\partial \text{ log pop density}}\right]\)
Directly interpreting statistical models is a full of traps that can mislead us. Working with predictions and marginal effects allows us to avoid many of these traps.
marginaleffects can be used with a huge array of models:
lm, glm)mgcv, gamm4)lme4, glmmTMB)brms)tidymodels, mlr3)On the one hand, we can avoid falling into many of the traps of interpreting model parameters.
On the other hand, this way of doing things brings the estimand into focus. We are being more explicit about the quantity we are trying to estimate–which is not tied to any particular model.
2,053 soccer players in the first male divisions of England, Germany, France, and Spain in the 2012–2013 season.
Outcome: number of red cards recieved for 146,028 player-referee dyads.
Covariates: player and referee traits, match statistics
“Are soccer players with dark skin tone more likely than those with light skin tone to receive red cards from referees?”
“Are soccer players with dark skin tone more likely than those with light skin tone to receive red cards from referees?”
“Thus, we argue that the results obtained in the CSI showed such a large variation because the 29 teams pursued (at least) four different research questions and therefore used different research designs.”
Controlled direct effect: \(E[Y(\text{dark}) - Y(\text{light}) | M = m]\); 486 parametric models
Causal (not statistical) assumptions needed to inform research design
Goals:
Statistics can feel like a thankless list of things that all have to be right or else nothing is right. - Not totally wrong, but leads to anxiety
Brauer, K. (2021). ” I’ll Finish It This Week” And Other Lies. arXiv preprint arXiv:2103.16574.
targetsPackage for reproducible research pipelines by Will Landau:
Encodes every step of your data analysis as a directed acyclic graph
Automatically detection of dependencies, which means:
targets will rerun (only) the parts that depend on the changed part.# _targets.R file
tar_source(files = "R") # load the R scripts in my dir
tar_option_set(packages = c("tidyverse", "marginaleffects")) # load any packages used
# define pipeline
list(
tar_target(data_file, "data/cognitive_flexibility.csv", format = "file"),
tar_target(data, read_csv(data_file)),
tar_target(model, fit_model(data)),
tar_target(marginal_effect, avg_comparisons(model, newdata = data, variables = "multilingual")),
tar_target(model_plot, plot_predictions(model, condition = list("multilingual")))
)▶ dispatched target data_file
● completed target data_file [0.402 seconds, 17.112 kilobytes]
▶ dispatched target data
Rows: 250 Columns: 4
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): multilingual
dbl (3): cognitive_flexibility, age, age_c
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
● completed target data [0.114 seconds, 5.687 kilobytes]
▶ dispatched target model
● completed target model [0.003 seconds, 17.378 kilobytes]
▶ dispatched target marginal_effect
● completed target marginal_effect [0.089 seconds, 41.914 kilobytes]
▶ dispatched target model_plot
● completed target model_plot [0.036 seconds, 253.351 kilobytes]
▶ ended pipeline [0.809 seconds]
library(targets)
source("R/functions.R")
# Dependencies for this pipeline
tar_option_set(packages = c(
"tidyverse",
"rethinking",
"rstan",
"patchwork",
"ggsci",
"RColorBrewer",
"cubelyr",
"proxy",
"foreach",
"doParallel",
"ggdist"
))
list(
# Extract raw data
tar_target(rdata_file,
"data/input-to-models.RData", format = "file"),
tar_target(footage_file,
"data/Study1_footage_details.xlsx"),
tar_target(infants_file,
"data/Infant_age.xlsx"),
tar_target(d_raw_time,
extract_raw_data(rdata_file, footage_file, infants_file, "time")),
## Generate data for Shiny dashboard #####
tar_target(d_dash,
dashboard_data(d_raw_time)),
# Data for time series model
tar_target(d_time,
time_data(d_raw_time)),
## Generate data for time-averaged model comparison
tar_target(d_avg,
time_data_avg(d_time)),
###### Time Averaged Analyses #########################
tar_target(model_compare,
avg_model_comparison(d_avg)),
###### Time Series Analyses ##########################
# Fit time series model
tar_target(fit_time,
fit_time_series(d_time, n_iter = 1000, n_chains = 8)),
# Make predictions for time series
tar_target(pred_time,
pred_species_pid_time(fit_time, d_time)),
# Make plots + export summaries of posterior predictions to .csv
# time series plots
tar_target(p_species_time_agent,
plot_species_time_AOI(pred_time, "Agent")),
tar_target(p_species_time_patient,
plot_species_time_AOI(pred_time, "Patient")),
tar_target(p_species_time_other,
plot_species_time_AOI(pred_time, "Other")),
tar_target(p_species_time_OR,
plot_species_time_OR(pred_time)),
tar_target(p_species_time_APO,
plot_species_time_OR_APO(pred_time)),
tar_target(p_species_time_AO,
plot_species_time_OR_AO(pred_time)),
# AOI size plots
tar_target(p_species_AOI_size,
plot_species_AOI_size(fit_time, d_time)),
# AP movement diff
tar_target(p_species_AP_md,
plot_species_AP_md(fit_time, d_time))
)targetsDynamic branching patterns, when there are too many targets to specify manually
Great support for parallel processing using future